LMFDB - Newform orbit 242.8.a.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [242,8,Mod(1,242)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("242.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(242, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 242 = 2 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	242.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,8,91] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$75.5971761672$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 22)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$

$=$

$ q + 8 q^{2} + 91 q^{3} + 64 q^{4} + 185 q^{5} + 728 q^{6} + 722 q^{7} + 512 q^{8} + 6094 q^{9} + 1480 q^{10} + 5824 q^{12} - 11020 q^{13} + 5776 q^{14} + 16835 q^{15} + 4096 q^{16} + 17210 q^{17} + 48752 q^{18}+ \cdots - 2418072 q^{98}+O(q^{100}) $

q + 8 * q^2 + 91 * q^3 + 64 * q^4 + 185 * q^5 + 728 * q^6 + 722 * q^7 + 512 * q^8 + 6094 * q^9 + 1480 * q^10 + 5824 * q^12 - 11020 * q^13 + 5776 * q^14 + 16835 * q^15 + 4096 * q^16 + 17210 * q^17 + 48752 * q^18 + 9288 * q^19 + 11840 * q^20 + 65702 * q^21 + 22971 * q^23 + 46592 * q^24 - 43900 * q^25 - 88160 * q^26 + 355537 * q^27 + 46208 * q^28 - 134272 * q^29 + 134680 * q^30 - 287765 * q^31 + 32768 * q^32 + 137680 * q^34 + 133570 * q^35 + 390016 * q^36 - 316397 * q^37 + 74304 * q^38 - 1002820 * q^39 + 94720 * q^40 + 335968 * q^41 + 525616 * q^42 + 858110 * q^43 + 1127390 * q^45 + 183768 * q^46 + 587680 * q^47 + 372736 * q^48 - 302259 * q^49 - 351200 * q^50 + 1566110 * q^51 - 705280 * q^52 - 244238 * q^53 + 2844296 * q^54 + 369664 * q^56 + 845208 * q^57 - 1074176 * q^58 - 163287 * q^59 + 1077440 * q^60 - 2297260 * q^61 - 2302120 * q^62 + 4399868 * q^63 + 262144 * q^64 - 2038700 * q^65 - 3428283 * q^67 + 1101440 * q^68 + 2090361 * q^69 + 1068560 * q^70 + 1542953 * q^71 + 3120128 * q^72 - 2216316 * q^73 - 2531176 * q^74 - 3994900 * q^75 + 594432 * q^76 - 8022560 * q^78 - 1526014 * q^79 + 757760 * q^80 + 19026289 * q^81 + 2687744 * q^82 - 1650370 * q^83 + 4204928 * q^84 + 3183850 * q^85 + 6864880 * q^86 - 12218752 * q^87 + 5760847 * q^89 + 9019120 * q^90 - 7956440 * q^91 + 1470144 * q^92 - 26186615 * q^93 + 4701440 * q^94 + 1718280 * q^95 + 2981888 * q^96 - 5750759 * q^97 - 2418072 * q^98

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1.1

8.00000

91.0000

64.0000

185.000

728.000

722.000

512.000

6094.00

1480.00

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$11$	$ -1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	242.8.a.f		1
11.b	odd	2	1		22.8.a.b	✓	1
33.d	even	2	1		198.8.a.d		1
44.c	even	2	1		176.8.a.a		1
55.d	odd	2	1		550.8.a.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
22.8.a.b	✓	1	11.b	odd	2	1
176.8.a.a		1	44.c	even	2	1
198.8.a.d		1	33.d	even	2	1
242.8.a.f		1	1.a	even	1	1	trivial
550.8.a.b		1	55.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{8}^{\mathrm{new}}(\Gamma_0(242))$:

$ T_{3} - 91 $

T3 - 91

$ T_{7} - 722 $

T7 - 722

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 8 $ T - 8
$3$	$ T - 91 $ T - 91
$5$	$ T - 185 $ T - 185
$7$	$ T - 722 $ T - 722
$11$	$ T $ T
$13$	$ T + 11020 $ T + 11020
$17$	$ T - 17210 $ T - 17210
$19$	$ T - 9288 $ T - 9288
$23$	$ T - 22971 $ T - 22971
$29$	$ T + 134272 $ T + 134272
$31$	$ T + 287765 $ T + 287765
$37$	$ T + 316397 $ T + 316397
$41$	$ T - 335968 $ T - 335968
$43$	$ T - 858110 $ T - 858110
$47$	$ T - 587680 $ T - 587680
$53$	$ T + 244238 $ T + 244238
$59$	$ T + 163287 $ T + 163287
$61$	$ T + 2297260 $ T + 2297260
$67$	$ T + 3428283 $ T + 3428283
$71$	$ T - 1542953 $ T - 1542953
$73$	$ T + 2216316 $ T + 2216316
$79$	$ T + 1526014 $ T + 1526014
$83$	$ T + 1650370 $ T + 1650370
$89$	$ T - 5760847 $ T - 5760847
$97$	$ T + 5750759 $ T + 5750759
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 242 = 2 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	242.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more